Trigonometry in the Right Triangle · Grade 10

What are sine, cosine and tangent?

All three are ratios between sides of a right triangle. Sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, and tangent is the opposite over the adjacent. Sine and cosine are always between zero and one, because the hypotenuse is the longest side.

Learning objectives

Three definitions

The sine of an angle is the length of the side opposite it divided by the hypotenuse. The cosine is the length of the adjacent side divided by the hypotenuse. The tangent is the opposite over the adjacent, and the hypotenuse plays no part in it at all.

All three are numbers, not lengths. They have no centimetres, because dividing a length by a length cancels the unit.

The safe way to remember is not an acronym but a fixed order: mark the angle, mark the hypotenuse, and only then ask which side is opposite it and which is beside it. Anyone who marks first almost never confuses sine with cosine.

Why sine is never greater than one

The hypotenuse is the longest side of a right triangle. So any leg divided by the hypotenuse is a number less than one, and sine and cosine must fall between zero and one for an acute angle.

Tangent has no such bound, because it is a ratio between two legs. As the angle approaches ninety degrees the adjacent side shrinks and the tangent grows without limit.

How the three connect

Tangent equals sine over cosine. You can see it directly: the hypotenuse appears in the numerator and the denominator of the two ratios and cancels.

So it is enough to remember two and derive the third — and this is also the identity that every trigonometric proof at five-unit level leans on later.

Worked examples

  1. In a right triangle the opposite side is 3, the adjacent 4 and the hypotenuse 5. Find all three ratios

    1. Sine: opposite over hypotenuse, that is 3 over 5
    2. Cosine: adjacent over hypotenuse, that is 4 over 5
    3. Tangent: opposite over adjacent, that is 3 over 4

    Answer: Sine 0.6, cosine 0.8, tangent 0.75

  2. Check that the tangent really equals sine over cosine

    1. From the previous exercise: sine 0.6 and cosine 0.8
    2. Divide: 0.6 over 0.8
    3. Compare with the tangent computed straight from the sides

    Answer: 0.75 — exactly the tangent computed directly

  3. Could the sine of an acute angle be 1.2?

    1. Sine is a leg over the hypotenuse
    2. The hypotenuse is longer than either leg
    3. So the quotient is less than one

    Answer: No. The sine of an acute angle is always between 0 and 1

Common mistakes

Swapping sine and cosine
Sine takes the side opposite the angle, cosine the one beside it. Both divide by the hypotenuse, which is what makes them easy to confuse — marking the angle on the diagram settles it.
Dividing by the hypotenuse when finding a tangent
The hypotenuse plays no part in the tangent. It is a ratio between the two legs, which is why it can be greater than one.
Giving the answer a unit
The ratios are numbers. Dividing centimetres by centimetres cancels the unit, so an answer of 0.6 centimetres is not right.

What to remember

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